Spectral radius and fractional matchings in graphs

Abstract

A {\it fractional matching} of a graph $G$ is a function $f$ giving each edge a number in $[0,1]$ so that ${\sum }_{e\in \Gamma (v)}f(e)\le 1$ for each $v\in V(G)$, where $\Gamma (v)$ is the set of edges incident to $v$. The {\it fractional matching number} of $G$, written ${\alpha }_{\ast }^{\prime }(G)$, is the maximum of ${\sum }_{e\in E(G)}f(e)$ over all fractional matchings $f$. Let $G$ be an $n$-vertex connected graph with minimum degree $d$, let ${\lambda }_1(G)$ be the largest eigenvalue of $G$, and let $k$ be a positive integer less than $n$. In this paper, we prove that if ${\lambda }_1(G)<d\sqrt{1+\frac{2k}{n-k}}$, then ${\alpha }_{\ast }^{\prime }(G)>\frac{n-k}{2}$. As a result, we prove ${\alpha }_{\ast }^{\prime }(G)\ge \frac{n{d}^2}{{\lambda }_1(G{)}^2+d^2}$, we characterize when equality holds in the bound.